The Hague Convention of 1899 states:
Art. 21. The obligations of belligerents with regard to the sick and
wounded are governed by the Geneva Convention of 22 August 1864,
subject to any modifications which may be introduced into it.
Art. 60. The Geneva Convention applies to sick and wounded interned
in neutral territory.
The Geneva Convention of 1864 states (my emphasis)
Article 1. Ambulances and military hospitals shall be recognized as neutral, and as such, protected and respected by the belligerents as long as they accommodate wounded and sick.
Neutrality shall end if the said ambulances or hospitals should be held by a military force.
Art. 2. Hospital and ambulance personnel, including the quarter-master's staff, the medical, administrative and transport services, and the chaplains, shall have the benefit of the same neutrality when on duty, and while there remain any wounded to be brought in or assisted.
Art. 6. .... Evacuation parties, and the personnel conducting them, shall be considered as being absolutely neutral.
Art. 7. A distinctive and uniform flag shall be adopted for hospitals, ambulances and evacuation parties. It should in all circumstances be accompanied by the national flag. An armlet may also be worn by personnel enjoying neutrality but its issue shall be left to the military authorities. Both flag and armlet shall bear a red cross on a white ground.
This clearly indicates that the traditional emblem of the International Red Cross shall designate medical personnel, and that medical personnel evacuating or treating wounded combatants are neutral.
Note however that it is medical personnel suitably identified and employed who are provided for - not just International Red Cross personnel.
The Armed Forces of the United States were for a long period governed by the Lieber Code of 1863 rather than international treaties:
Art. 116. Honorable belligerents often request that the hospitals within the territory of the enemy may be designated, so that they may be spared. An honorable belligerent allows himself to be guided by flags or signals of protection as much as the contingencies and the necessities of the fight will permit.
The Lieber Code additionally goes into great detail surrounding the proper handling of Flags of Truce arranged between opposing commanders. But because it was a unilateral declaration of rules of engagement for Union forces, it is constrained to speak in more general terms than subsequent international treaties.